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User: Ethan Beihl

Ethan Beihl has authored 3 sequences.

A278823 4-Portolan numbers: number of regions formed by n-secting the angles of a square.

1, 4, 29, 32, 93, 84, 189, 188, 321, 316, 489, 460, 693, 676, 933, 916, 1205, 1180, 1505, 1496, 1849, 1836, 2229, 2188, 2645, 2616, 3097, 3060, 3577, 3536, 4089, 4064, 4645, 4604, 5237, 5176, 5857, 5808, 6513, 6472, 7201, 7160, 7933, 7900, 8693, 8648, 9497

Offset: 1

Ethan Beihl, Nov 28 2016

nonn

m-Portolan numbers for m>3 (especially m even) are more difficult than m=3 (A277402) because Ceva's theorem doesn't immediately give us a condition for redundant intersections. The values for n <= 23 were found by brute force in Mathematica, as follows:
1. Solve for the coordinates of all intersections between lines within the square, recording multiplicity.
2. Use an elementary Euler's-formula method as in Poonen and Rubinstein 1998 to turn the intersection-count into a region-count.

			For n=3, the 4*(3-1) = 8 lines intersect to make 12 triangles, 8 kites, 8 irregular quadrilaterals, and an octagon in the middle. The total number of regions a(3) is therefore 12+8+8+1 = 29.

Lars Blomberg, Table of n, a(n) for n = 1..500
Ethan Beihl, Pictures for some small n
Lars Blomberg, Coloured illustration for n=4
Lars Blomberg, Coloured illustration for n=5
Lars Blomberg, Coloured illustration for n=64
Lars Blomberg, Coloured illustration for n=65
B. Poonen and M. Rubinstein (1998) The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:10.1137/S0895480195281246, arXiv:math.MG/9508209 (typos corrected)

3-Portolan numbers (equilateral triangle): A277402.
n-sected sides (not angles): A108914.
Cf. A277402, A335526 (vertices), A335527 (edges), A335528 (ngons).

For n = 2k - 1, a(n) is close to 18k^2 - 26k + 9. For n = 2k, a(n) is close to 18k^2 - 26k + 12. The residuals are related to the structure of redundant intersections in the figure.

a(24) and beyond from Lars Blomberg, Jun 12 2020

A277402 "3-Portolan numbers": number of regions formed by n-secting the angles of an equilateral triangle.

Original entry on oeis.org

1, 6, 19, 30, 61, 78, 127, 150, 217, 234, 331, 366, 469, 510, 631, 678, 817, 870, 1027, 1074, 1261, 1326, 1519, 1590, 1801, 1878, 2107, 2190, 2437, 2514, 2791, 2886, 3169, 3270, 3571, 3678, 3997, 4110, 4447, 4554, 4921, 5046, 5419, 5550, 5941, 6078, 6487, 6630, 7057, 7194

Offset: 1

Ethan Beihl, Oct 13 2016

nonn

I like the name "portolan numbers": cf. the rhumbline designs on medieval maps, constructed in a similar way.
The regions can be counted using an adaptation of Smiley and Wick's method in A092098: count regions assuming there are no redundant intersections, then subtract the number of regions that Ceva's Theorem says must vanish.
Off-diagonal redundant intersections occur for triples of integers 1 <= i, j, k < floor(n/2)-1 such that M(i)*M(j) = M(k), where M(x) is the ratio (sin(Pi(n-x)/(3n)))/(sin(Pi*x/(3n))). In the case 10|n, this corresponds to the charming identity (sin(7*Pi/30)*sin(8*Pi/30))/(sin(3*Pi/30)*sin(2*Pi/30)) = sin(9*Pi/30)/sin(1*Pi/30).
Differs from A092098 (which counts regions when *sides*, not angles, are n-sected) for the first time at the tenth term.
The above equation has solutions if and only if 10|n. This can be shown by rewriting the equation in exponential form, and using facts about vanishing sums of roots of unity to narrow the possibilities for n. (See Conway and Jones, 1976.) This is computationally feasible because A273096(6) = 1. - Ethan Beihl, Nov 26 2016

			For n=3, a(n) gives the 19 regions formed by the intersection of 3*2 lines: 3 pentagons, 3 quadrilaterals, 12 triangles, and 1 big central hexagon.

Lars Blomberg, Table of n, a(n) for n = 1..500
Lars Blomberg, Coloured illustration for n=3
Lars Blomberg, Coloured illustration for n=4
Lars Blomberg, Coloured illustration for n=19
Lars Blomberg, Coloured illustration for n=20
J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arithmetica 30(3), 229-240 (1976).
Wikipedia, Rhumbline network

Cf. A092098, A335411 (vertices), A335412 (edges), A335413 (ngons).

regions[n_]:=
If[Mod[n,2]==0, 3n^2-6n+6, 3n^2-3n+1]-
  6*Length@
    Select[
     Flatten@
      With[
       {b=N@
          Table[
             1/2 - (Sqrt[3]/2)Tan[(60Degree/n)(n/2-i)],
             {i, 1, Floor[n/2]- 1}
             ]},
       Table[
        Abs[(1-b[[k]])b[[l]]b[[j]] - b[[k]](1-b[[l]])(1-b[[j]])],
        {j, 1, Floor[n/2]-1},
        {k, 1, Floor[n/2]-1},
        {l, 1, Floor[n/2]-1}]
       ],
     Chop@#==0&]

Empirical g.f.: x*(1 + 5*x + 12*x^2 + 6*x^3 + 18*x^4 + 6*x^5 + 18*x^6 + 6*x^7 + 18*x^8 - 6*x^9 + 29*x^10 + 13*x^11 - 6*x^12) / ((1-x)^3*(1+x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Oct 14 2016
Empirically for 12 < n <= 500: a(n) = a(n-2) + a(n-10) - a(n-12) + 120. - Lars Blomberg, Jun 08 2020
Empirical: a(2*k + 1) = 6*k*(2*k + 1) + 1, for k >= 0. - Ivan N. Ianakiev, Jun 27 2020
Empirical: 10*a(n) = 30*n^2 -45*n +23 +13*(-1)^n -15*(-1)^n*n - 24*b(n) where b(n) is the 10-periodic sequence 4, 0, -1, 0, -1, 0, -1, 0, -1, 0, 4, 0 .... of offset 0. - R. J. Mathar, Jul 05 2020

A166361 Scale degrees of the roots of chords in a traditional "twelve-bar blues" in Western music.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 1, 5, 4, 1, 1

Offset: 1

Ethan Beihl, Oct 12 2009

			In key of C: C7 C7 C7 C7 F7 F7 C7 C7 G7 F7 C7 C7

Tonic * 4; Subdominant * 2; Tonic * 2; Dominant; Subdominant; Tonic * 2

Keyword:fini,full added by R. J. Mathar, Oct 14 2009

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User: Ethan Beihl

A278823 4-Portolan numbers: number of regions formed by n-secting the angles of a square.

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A277402 "3-Portolan numbers": number of regions formed by n-secting the angles of an equilateral triangle.

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A166361 Scale degrees of the roots of chords in a traditional "twelve-bar blues" in Western music.

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Examples

Formula

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